Perpendicular Line Equation: Find It Easily!

Let's dive into finding the equation of a line that's perpendicular to a given line and passes through a specific point. It sounds a bit complex, but trust me, it's totally manageable! We're going to break it down step by step so you can ace this type of problem every time.

Understanding Perpendicular Lines

When we talk about perpendicular lines, we're talking about lines that intersect at a right angle (90 degrees). The key to identifying perpendicular lines lies in their slopes. If two lines are perpendicular, the product of their slopes is -1. In simpler terms, the slope of one line is the negative reciprocal of the slope of the other. This is a fundamental concept that we'll use throughout this problem. The original line is given by the equation $y = -0.3x + 6$. This equation is in slope-intercept form, which is $y = mx + b$, where m represents the slope and b represents the y-intercept. In our case, the slope of the given line is -0.3, which can also be written as -3/10. To find the slope of a line perpendicular to this one, we need to find the negative reciprocal of -3/10. First, we take the reciprocal, which means we flip the fraction to get -10/3. Then, we take the negative of that, which gives us 10/3. So, the slope of the line perpendicular to $y = -0.3x + 6$ is 10/3. Remember, the relationship between the slopes of perpendicular lines is crucial, and understanding this concept is the first step to solving this problem. In summary, visualizing these lines can be super helpful. Imagine one line going downwards slightly as you move from left to right (that's our original line with a negative slope of -0.3), and another line intersecting it at a perfect right angle, going upwards much steeper (that’s the perpendicular line we're trying to find, with a positive slope of 10/3). Grasping this visual will make the math much more intuitive.

Finding the Equation

Now that we know the slope of our perpendicular line, which is 10/3, we can start building its equation. We'll use the point-slope form of a line, which is: $y - y_1 = m(x - x_1)$, where: * m is the slope of the line. * $(x_1, y_1)$ is a point on the line. We are given the point (3, -8) that the line passes through. So, $x_1 = 3$ and $y_1 = -8$. Plugging these values and the slope (m = 10/3) into the point-slope form, we get: $y - (-8) = (10/3)(x - 3)$. Simplifying this, we have: $y + 8 = (10/3)(x - 3)$. Next, we distribute the 10/3 on the right side of the equation: $y + 8 = (10/3)x - (10/3)(3)$. $y + 8 = (10/3)x - 10$. Now, we want to isolate y to get the equation in slope-intercept form (y = mx + b). To do this, we subtract 8 from both sides of the equation: $y = (10/3)x - 10 - 8$. $y = (10/3)x - 18$. So, the equation of the line that is perpendicular to $y = -0.3x + 6$ and passes through the point (3, -8) is $y = (10/3)x - 18$. This equation tells us everything we need to know about the line: its slope (10/3) and its y-intercept (-18). Always double-check your work by ensuring the new slope is the negative reciprocal of the original slope, and that the point (3,-8) satisfies the equation. Subsituting x = 3 into the equation yields y = (10/3)*3 - 18 = 10 - 18 = -8, which confirms our result.

Verifying the Solution

To make sure our solution is correct, let's verify that the line $y = (10/3)x - 18$ indeed passes through the point (3, -8) and is perpendicular to the line $y = -0.3x + 6$. First, let's check if the point (3, -8) satisfies the equation $y = (10/3)x - 18$. Substitute x = 3 into the equation: $y = (10/3)(3) - 18$. $y = 10 - 18$. $y = -8$. Since the y-value we obtained is -8, the point (3, -8) lies on the line $y = (10/3)x - 18$. Next, let's verify that the lines are perpendicular. The slope of the given line $y = -0.3x + 6$ is -0.3 or -3/10. The slope of the line we found, $y = (10/3)x - 18$, is 10/3. To check if the lines are perpendicular, we multiply their slopes: $(-3/10) * (10/3) = -1$. Since the product of the slopes is -1, the lines are indeed perpendicular. Therefore, the equation $y = (10/3)x - 18$ is the correct equation of the line that is perpendicular to $y = -0.3x + 6$ and passes through the point (3, -8). This verification step is crucial because it ensures that we haven't made any mistakes in our calculations. By plugging in the point and checking the slopes, we can be confident that our solution is accurate. And that's how you find the equation, guys!

Alternative Methods

While we've used the point-slope form to solve this problem, there are other methods you could use. Let's explore a couple of alternatives to give you more tools in your problem-solving toolkit. One alternative method involves directly using the slope-intercept form (y = mx + b) after finding the perpendicular slope. We know the perpendicular slope is 10/3, so our equation looks like: $y = (10/3)x + b$. Now, we need to find the y-intercept (b). We can do this by plugging in the point (3, -8) into the equation: $-8 = (10/3)(3) + b$. $-8 = 10 + b$. Subtracting 10 from both sides gives us: $b = -18$. So, our equation is $y = (10/3)x - 18$, which matches our previous result. This method is often quicker for those comfortable manipulating equations. Another approach involves a more conceptual understanding of transformations. Since we know the slope of the perpendicular line and a point it passes through, we can think about shifting the original line until it passes through the given point. This method requires a solid understanding of how changing the y-intercept affects the position of the line. While these alternative methods can be useful, the point-slope form is generally the most straightforward and reliable, especially when dealing with points that aren't easily visualized on the y-axis. Experimenting with different methods will help you develop a deeper understanding of linear equations and improve your problem-solving skills. Ultimately, the best method is the one that you understand the most thoroughly and can apply consistently and accurately.

Common Mistakes to Avoid

When working with perpendicular lines and equations, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer every time. One frequent mistake is forgetting to take the negative reciprocal of the slope. Remember, perpendicular lines have slopes that are negative reciprocals of each other. For example, if the original slope is 2/3, the perpendicular slope is -3/2. It's easy to remember to flip the fraction but forget to change the sign, or vice versa. Always double-check that you've done both! Another common error is incorrectly applying the point-slope form. Make sure you substitute the x and y values of the given point into the correct places in the formula: $y - y_1 = m(x - x_1)$. It's also crucial to pay attention to signs, especially when dealing with negative coordinates. A simple sign error can throw off your entire calculation. Another pitfall is not simplifying the equation correctly after substituting the values. Be careful when distributing and combining like terms. Take your time and double-check each step to avoid arithmetic errors. Finally, always verify your solution by plugging the given point into the final equation and checking if the product of the slopes is -1. This will help you catch any mistakes you might have made along the way. By being mindful of these common errors and taking the time to check your work, you can increase your chances of getting the correct answer and build confidence in your problem-solving abilities. Remember, practice makes perfect, so keep working at it!

Practice Problems

To really solidify your understanding of finding equations of perpendicular lines, let's go through a few practice problems. Working through these examples will help you apply the concepts we've discussed and identify any areas where you might need further review. Problem 1: Find the equation of a line that is perpendicular to $y = 2x - 3$ and passes through the point (4, 1). First, find the slope of the perpendicular line. The slope of the given line is 2, so the slope of the perpendicular line is -1/2. Next, use the point-slope form: $y - 1 = (-1/2)(x - 4)$. Simplify to get: $y = (-1/2)x + 3$. Problem 2: Find the equation of a line that is perpendicular to $y = (-3/4)x + 5$ and passes through the point (-2, 6). The slope of the given line is -3/4, so the slope of the perpendicular line is 4/3. Use the point-slope form: $y - 6 = (4/3)(x - (-2))$. Simplify to get: $y = (4/3)x + 26/3$. Problem 3: Find the equation of a line that is perpendicular to $y = (1/5)x + 2$ and passes through the point (0, -3). The slope of the given line is 1/5, so the slope of the perpendicular line is -5. Use the point-slope form: $y - (-3) = -5(x - 0)$. Simplify to get: $y = -5x - 3$. By working through these practice problems, you'll become more comfortable with the process of finding equations of perpendicular lines. Remember to always double-check your work and verify your solution to ensure accuracy. The more you practice, the more confident you'll become in your ability to solve these types of problems.